The Annals of Applied Probability

No-feedback card guessing for dovetail shuffles

Mihai Ciucu

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We consider the following problem. A deck of $2n$ cards labeled consecutively from 1 on top to $2n$ on bottom is face down on the table. The deck is given k dovetail shuffles and placed back on the table, face down. A guesser tries to guess at the cards one at a time, starting from top. The identity of the card guessed at is not revealed, nor is the guesser told whether a particular guess was correct or not. The goal is to maximize the number of correct guesses. We show that, for $k \geq 2 \log_2 (2n) + 1$, the best strategy is to guess card 1 for the first half of the deck and card $2n$ for the second half. This result can be interpreted as indicating that it suffices to perform the order of $\log_2(2n)$ shuffles to obtain a well-mixed deck, a fact proved by Bayer and Diaconis. We also show that if $k = c \log_2 (2n)$ with $1 < c < 2$, then the above guessing strategy is not the best.

Article information

Ann. Appl. Probab., Volume 8, Number 4 (1998), 1251-1269.

First available in Project Euclid: 9 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Card guessing dovetail shuffle riffle shuffle


Ciucu, Mihai. No-feedback card guessing for dovetail shuffles. Ann. Appl. Probab. 8 (1998), no. 4, 1251--1269. doi:10.1214/aoap/1028903379.

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